The spectral localizer for even index pairings

TL;DR

The paper introduces the spectral localizer, a finite-dimensional matrix that makes computing even index pairings numerically accessible, bridging abstract index theory with practical computation.

Contribution

It constructs the spectral localizer, a selfadjoint invertible matrix, enabling numerical evaluation of even index pairings in non-commutative geometry.

Findings

Spectral localizer's signature equals the even index pairing

Provides a practical method for computing index invariants

Uses fuzzy spheres in the index-theoretic proof

Abstract

Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a $K_0$-class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite dimensional selfadjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the even index pairing. This makes the invariant numerically accessible. The index-theoretic proof heavily uses fuzzy spheres.